The construction of kernel density estimates \(f^{(t,m)}_i\)  corresponding to spherical shell \(t \in \lbrace 1,\ldots ,\tau \rbrace \)  from MRI sequence \(m \in M = \lbrace T1,T1Gd,T2,FLAIR\rbrace \)  for subject \(i\)  are described in Algorithm REF . These kernel density estimates are computed using the density function in R using the default bandwidth selection method, also commonly referred to as Silverman's rule, which is an optimal bandwidth for estimating Gaussian densities [1]}.
[!h]
Construction of layer-wise kernel density estimates
[1]
Input: MRI scans from \(M\)  sequences and corresponding tumor segmentation for \(n\)  subjects.
Output: PDFs corresponding to \(\tau \)  layers from \(M\)  imaging sequences for \(n\)  subjects.
each subject \(i=1,\ldots ,n\) 
Define \(N_i\)  as the tumor volume (number of tumor voxels).
each MRI sequence \(m \in M\) 
each layer \(t=1,\ldots ,\tau \) 
\(t=1\) 
[t]@X@
Define \(S^{(1)}_i\)  as the sphere with \(\mbox{{$\mathbf {c}$}}_i\)  as its center and radius \(r^{(1)}_i = \inf \limits _{r \in \mathbb {R}^+} r\)  such that \(r\)  is the radius of all spheres centered at \(\mbox{{$\mathbf {c}$}}_i\)  and containing at least \(N_i/\tau \)  voxels.


[t]@X@
Construct \(f^{(1,m)}_i\)  as the kernel density estimate based on the voxels in sequence \(m\)  belonging to the sphere \(S^{(1)}_i\) .


\(t>1\) 
[t]@X@
Define the spherical shell \(D^{(t)}_i = S^{(t)}_i\backslash S^{(t-1)}_i\) , where \(S^{(t)}_i\)  is the sphere with center \(\mbox{{$\mathbf {c}$}}_i\)  and radius \(r^{(t)}_i = \inf \limits _{r \in \mathbb {R}^+} r\)  such that \(r\)  is the radius of all spheres centered at \(\mbox{{$\mathbf {c}$}}_i\)  and containing at least \(tN_i/\tau \)  voxels.


[t]@X@
Construct \(f^{(t,m)}_i\)  as the kernel density estimate based on the voxels in sequence \(m\)  belonging to the spherical shell \(D^{(t)}_i\) 

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